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Μαθηματική Ανάσυρση
Ανάσυρσις Pullback thumb|300px| [[Μαθηματική Ανάσυρση |Ανάσυρση (pullback) ]] thumb|300px| [[Μαθηματική Ανάσυρση |Ανάσυρση (pullback) Προώθηση (pushforward) ]] thumb|300px| [[Μαθηματική Ανάσυρση ]] thumb|300px| [[Μαθηματική Ανάσυρση ]] thumb|300px| [[Μαθηματική Ανάσυρση Μαθηματική Προώθηση ]] - Μία διαδικασία. Ετυμολογία Η ονομασία "Ανάσυρση" σχετίζεται ετυμολογικά με την λέξη "σάρωση". Ορισμός Στο σχήμα (1) we consider two manifolds M and N, (possibly of different dimension), with coordinate systems xμ and yν, respectively. We imagine that we have a map: : \phi \colon M \rightarrow N and a function: : f\colon N \rightarrow R . It is obvious that we can compose the map φ'' with the function''f to construct a map: : ( f \circ \phi) \colon M \rightarrow R , which is simply a function on manifold M. Such a construction is, sufficiently, useful that it gets its own name; we define the pullback of the function f'' by the map ''φ, denoted \phi_*f , by : \phi_*f = f \circ \phi The name makes sense, since we think of \phi_*f as "pulling back" the function f from manifold N to manifold M. Εισαγωγή In mathematics, a pullback is either of two different, but related processes: precomposition and fibre-product. Its "dual" is pushforward measure. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: in simple terms, a function f'' of a variable ''y, where y'' itself is a function of another variable ''x, may be written as a function of x''. This is the pullback of ''f by the function y''. : f(y(x)) \equiv g(x) \, It is such a fundamental process, that it is often passed over without mention, for instance in elementary calculus: this is sometimes called ''omitting pullbacks, and pervades areas as diverse as fluid mechanics and differential geometry. However, it is not just functions that can be "pulled back" in this sense. Pullbacks can be applied to many other objects such as differential forms and their cohomology classes. See: *Pullback (differential geometry) *Pullback (cohomology) Fibre-product The notion of pullback as a fibre-product ultimately leads to the very general idea of a categorical pullback, but it has important special cases: * inverse image (and pullback) sheaves in algebraic geometry, and * pullback bundles in algebraic topology and differential geometry. The pullback bundle is perhaps the simplest example that bridges * the notion of a pullback as precomposition, and * the notion of a pullback as a cartesian square. In that example, the base space of a fiber bundle is pulled back, in the sense of precomposition, above. The fibers then travel along with the points in the base space at which they are anchored: the resulting new pullback bundle looks locally like a cartesian product of the new base space, and the (unchanged) fiber. The pullback bundle then has two projections: *one to the base space, *the other to the fiber; the product of the two becomes coherent when treated as a fiber product. See: *Pullback (category theory) *Inverse image sheaf *Pullback bundle *Fibred category Functional analysis When the pullback is studied as an operator acting on function spaces, it becomes a linear operator, and is known as the composition operator. Its adjoint is the push-forward, or, in the context of functional analysis, the transfer operator. Relationship The relation between the two notions of pullback can perhaps best be illustrated by sections of fibre bundles: if s'' is a section of a fibre bundle ''E over N'', and ''f is a map from M'' to ''N, then the pullback (precomposition) f^* s=s\circ f of s'' with ''f is a section of the pullback (fibre-product) bundle f''*''E over M. Υποσημειώσεις Εσωτερική Αρθρογραφία * Μαθηματική Ανάσυρση * Μαθηματική Προώθηση * Inverse image functor Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *ned.ipac.caltech.edu Πλήρης εξήγηση *pullback bundle * mathphysicsbook.com * iue.tuwien.ac.at * pullback and putforthward * Lecture Notes on General Relativity, Carroll